From Wikipedia, the free encyclopedia
Exponential martingale associated to sum of iid variables
In probability theory, Wald's martingale is the name sometimes given to a martingale used to study sums of i.i.d. random variables. It is named after the mathematician Abraham Wald, who used these ideas in a series of influential publications.[1][2][3]
Wald's martingale can be seen as discrete-time equivalent of the Doléans-Dade exponential.
Formal statement
[edit]Let be a sequence of i.i.d. random variables whose moment generating function
is finite for some
, and let
, with
. Then, the process
defined by
is a martingale known as Wald's martingale.[4] In particular, for all
.
See also
[edit]Notes
[edit]- ^ Wald, Abraham (1944). "On cumulative sums of random variables". Ann. Math. Stat. 15 (3): 283–296. doi:10.1214/aoms/1177731235.
- ^ Wald, Abraham (1945). "Sequential tests of statistical hypotheses". Ann. Math. Stat. 16 (2): 117–186. doi:10.1214/aoms/1177731118.
- ^ Wald, Abraham (1945). Sequential analysis (1st ed.). John Wiley and Sons.
- ^ Gamarnik, David (2013). "Advanced Stochastic Processes, Lecture 10". MIT OpenCourseWare. Retrieved 24 June 2023.
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